Riemann curvature of Regge metrics: from angle defects to finite element approximation
Date:
The slides can be found here.
Abstract:
Regge metrics are piecewise smooth Riemannian metrics whose induced metrics agree on common element faces. This tangential-tangential continuity allows neighboring elements to be glued isometrically, although the full metric tensor may jump across interfaces. For piecewise constant metrics, classical Regge calculus describes curvature through angle defects. Higher-order metrics also carry curvature in element interiors and along their faces.
In this talk, we start from the two-dimensional Gauss-Bonnet formula and present a construction of the full Riemann curvature for Regge metrics in arbitrary dimension. The resulting functional combines elementwise curvature, jumps of the second fundamental form, and angle defects at codimension-two interfaces. After identifying a common test space, we establish quantitative convergence estimates in a negative Sobolev norm for suitable Regge finite element approximations of sufficiently smooth metrics. We present numerical examples performed in the open-source finite element software NGSolve (www.ngsolve.org).
This is joint work with Jay Gopalakrishnan, Joachim Schöberl, and Max Wardetzky.